I think that you want an UpdateCursor to change the field value rather than inserting a new feature, which is what the InsertCursor does.
You mention that you want to "find out which protected areas (output_PA_clip) fall into a planning unit (spatialjoin_fc)..." The way it is in your code, it is backwards. It's saying select if the spatial_join feature has ITS center in the output_PA_clip. You may be missing valid selections and also getting false positives. Do they just have to intersect or do they need to have their center within the planning unit?
You may be able to get away with only one cursor, iterating over the selected items and updating them to true.
Edit: You also have == in your row[0] assignment that should be one =.
selectedFeatures = arcpy.SelectLayerByLocation_management(spatialjoin_fc, "INTERSECT", output_PA_clip)
with arcpy.da.UpdateCursor(selectedFeatures, ["lock_out"]) as cursor:
for row in cursor:
row[0] = 'TRUE'
cursor.updateRow(row)